You must have at least one "symbol" font (e.g. Symbol or Euclid Symbol) and at least one "extra" font (e.g. MT Extra or Euclid Extra)

Oddly enough, the second time launching MathType, the error would no longer appear. Since the Guest account deletes all files when you log out, though, the error would appear again the next time the Guest account logged in.

What I tried that didn't work

Search for the solution online (no useful results).

Going into Font Book to check for and resolve enabled duplicate fonts.

Log in, launch MathType once to get the error and then launch it again and then copy the Guest profile to the User Template.

Repair permissions.

Try another computer to see if the problem can be duplicated (it can).

The actual solution

Using an admin account, launch up Font Book.app and in the search filter, search for MT Extra. Find the duplicate and remove (delete) it.

In my case (using Microsoft Office 2011), one was in /Library/Fonts/Microsoft and the other was in /Library/Fonts.

Same deal for Symbol, except I found one in /System/Library/Fonts and the other in /Library/Fonts

Note: in this case of this MathType error, it is not enough to simply disable the duplicate fonts. You actually have to remove the duplicates completely, or MathType will still error out the first time it's launched in the Guest account.